Lecture arc
Survey astronomy is an inverse problem: the catalogue is filtered by the instrument and by the scientist's cuts. Start by asking students what the telescope actually measures. Then introduce likelihoods, priors, and selection functions; delay the equation until the geometry and assumptions are visible. Use the visual as an argument: students should identify what is data, what is a model, and what is inferred.
The lecture should return to the one-dimensional worked calculation and ask whether the answer is stable under a plausible change in distance, tracer selection, or model family.
Terms Developed in Context
Likelihood
A model for the probability of the observed data as a function of parameters, \(p(D|\theta)\). It encodes the measurement uncertainties used to compare Galactic models with survey data. For independent Gaussian rotation points it is related to a chi-square residual sum. In this lecture, use this term while reasoning about likelihoods, priors, and selection functions, and distinguish the measured quantity from the model-dependent quantity in \(p(\theta|D)\propto p(D|\theta)p(\theta)\). A convenient likelihood can be precise yet wrong if errors are correlated or non-Gaussian. This boundary is important when comparing tracers or carrying the result into the worked example.
Prior
A probability distribution for parameters before conditioning on the current data. It regularizes underconstrained distance and Galactic-structure inferences and makes assumptions explicit. Bayes' theorem combines it with the likelihood to form \(p(\theta|D)\). In this lecture, use this term while reasoning about likelihoods, priors, and selection functions, and distinguish the measured quantity from the model-dependent quantity in \(p(\theta|D)\propto p(D|\theta)p(\theta)\). A prior is not an additional observation, and an apparently weak prior can matter with sparse data. This boundary is important when comparing tracers or carrying the result into the worked example.
Posterior
The probability distribution of model parameters after combining likelihood and prior. It summarizes parameter estimates, uncertainty, and covariance for the adopted model. Posterior predictive checks connect inferred parameters back to observable catalogues. In this lecture, use this term while reasoning about likelihoods, priors, and selection functions, and distinguish the measured quantity from the model-dependent quantity in \(p(\theta|D)\propto p(D|\theta)p(\theta)\). A narrow posterior does not protect against a misspecified model or selection function. This boundary is important when comparing tracers or carrying the result into the worked example.
Selection function
The probability that an astronomical source is detected and retained as a function of its properties. It is part of the data-generating process and must enter population inference. The predicted underlying population is multiplied by selection before comparison with a catalogue. In this lecture, use this term while reasoning about likelihoods, priors, and selection functions, and distinguish the measured quantity from the model-dependent quantity in \(p(\theta|D)\propto p(D|\theta)p(\theta)\). Cuts applied by the analyst are part of the selection, not harmless bookkeeping. This boundary is important when comparing tracers or carrying the result into the worked example.
Posterior predictive check
A comparison between data simulated from the fitted posterior model and the actual observations. It tests whether the model reproduces features beyond the fitted parameter summary. Residual distributions, trends, and replicated catalogues reveal model-data mismatch. In this lecture, use this term while reasoning about likelihoods, priors, and selection functions, and distinguish the measured quantity from the model-dependent quantity in \(p(\theta|D)\propto p(D|\theta)p(\theta)\). Passing selected checks does not prove the model unique or physically true. This boundary is important when comparing tracers or carrying the result into the worked example.